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13. Hemmatnezhad, M., Ansari, R., Rahimi, G.H.: Large-amplitude free vibrations of functionally
graded beams by means of a finite element formulation. Appl. Math. Model. 37, 8495–8504
(2013)
14. Kien, N.D.: Large displacement response of tapered cantilever beams made of axially functionally graded material. Compos. Part B 55, 298–309 (2013)
15. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35(1), 412–425 (2011)
16. Shahba, A., Attarnejad, R., Hajilar, S.: Free vibration and stability of axially functionally
graded tapered Euler-Bernoulli beams. Shock Vib. 18(5), 683–696 (2011)
17. Shahba, A., Attarnejad, R., Tavanaie, M.M., Hajilar, S.: Free vibration and stability analysis
of axially functionally graded tapered Timoshenko beams with classical and non-classical
boundary conditions. Compos. Part B: Eng. 42(4), 0801–0808 (2011)
18. Ke, L.L., Yang, J., Kitipornchai, S.: An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45, 743–752 (2010)
19. Akgoz, B., Civalek, O.: Free vibration analysis of axially functionally graded tapered
Bernoulli-Euler microbeams based on the modified couple stress theory. Compos. Struct.
98, 314–322 (2013)
20. Banerjee, J.R., Su, H., Jackson, D.R.: Free vibration of rotating tapered beams using the
dynamic stiffness method. J. Sound Vib. 298(4–5), 1034–1054 (2006)
21. Hsu, J.-C., Lai, H.-Y., Chen, C.-K.: Free vibration of nonuniform Euler-Bernoulli beams with
general elastically end constraints using Adomian modified decomposition method. J. Sound
Vib. 318(4–5), 965–981 (2008)
22. Huang, Y., Li, X.-F.: A new approach for free vibration of axially functionally graded beams
with non-uniform crosssection. J. Sound Vib. 329(11), 2291–2303 (2010)
23. Li, X.-F., Kang, Y.-A., Wu, J.-X.: Exact frequency equations of free vibration of exponentially
functionally graded beams. Appl. Acoust. 74(3), 413–420 (2013)
24. Mao, Q., Pietrzko, S.: Free vibration analysis of a type of tapered beams by using Adomian
decomposition method. Appl. Math. Comput. 219(6), 3264–271 (2012)
25. Ozgumus, O.-O., Kaya, M.O.: Flapwise bending vibration analysis of a rotating doubletapered Timoshenko beam. Arch. Appl. Mech. 78(5), 379–392 (2008)
26. Rajasekaran, S.: Free vibration of centrifugally stiffened axially functionally graded tapered
Timoshenko beams using differential transformation and quadrature methods. Appl. Math.
Model. 37(6), 4440–4463 (2013)
27. Rajasekaran, S., Norouzzadeh, T.E.: Free vibration analysis of axially functionally graded
tapered Timoshenko beams using differential transformation element method and differential
quadrature element method of lowest-order. Meccan. 49(4), 995–1009 (2014)
28. Yardimoglu, B.: Vibration analysis of rotating tapered Timoshenko beams by a new finite
element model. Shock Vib. 13(2), 117–126 (2006)
29. Baghani, M., Mazaheri, H., Salarieh, H.: Analysis of large amplitude free vibrations of
clamped tapered beams on a nonlinear elastic foundation. Appl. Math. Model. 38(3), 1176–
1186 (2013)
30. Huang, Y., Yang, L.E., Luo, Q.Z.: Free vibration of axially functionally graded Timoshenko
beams with non-uniform cross-section. Compos. Part B: Eng. 45, 1493–1498 (2013)
31. Shahba, A., Attarnejad, R., Hajilar, S.: A Mechanical-Based Solution for Axially Functionally
Graded Tapered Euler-Bernoulli Beams. Mech. Adv. Mater. Struct. 20, 696–707 (2012)
32. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent
behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)
33. Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1766 (2010)
34. Lazopoulos, K.A., Lazopoulos, A.K.: Bending and buckling of thin strain gradient elastic
beams. Eur. J. Mech. A-Sol. 29, 837–843 (2010)
397
12. Elishakoff, I., Johnson, V.: Apparently the first closed-form solution of vibrating inhomogeneous beam with a tip mass. J. Sound Vib. 286(4–5), 1057–1066 (2009)
13. Hemmatnezhad, M., Ansari, R., Rahimi, G.H.: Large-amplitude free vibrations of functionally
graded beams by means of a finite element formulation. Appl. Math. Model. 37, 8495–8504
(2013)
14. Kien, N.D.: Large displacement response of tapered cantilever beams made of axially functionally graded material. Compos. Part B 55, 298–309 (2013)
15. Alshorbagy, A.E., Eltaher, M.A., Mahmoud, F.F.: Free vibration characteristics of a functionally graded beam by finite element method. Appl. Math. Model. 35(1), 412–425 (2011)
16. Shahba, A., Attarnejad, R., Hajilar, S.: Free vibration and stability of axially functionally
graded tapered Euler-Bernoulli beams. Shock Vib. 18(5), 683–696 (2011)
17. Shahba, A., Attarnejad, R., Tavanaie, M.M., Hajilar, S.: Free vibration and stability analysis
of axially functionally graded tapered Timoshenko beams with classical and non-classical
boundary conditions. Compos. Part B: Eng. 42(4), 0801–0808 (2011)
18. Ke, L.L., Yang, J., Kitipornchai, S.: An analytical study on the nonlinear vibration of functionally graded beams. Meccanica 45, 743–752 (2010)
19. Akgoz, B., Civalek, O.: Free vibration analysis of axially functionally graded tapered
Bernoulli-Euler microbeams based on the modified couple stress theory. Compos. Struct.
98, 314–322 (2013)
20. Banerjee, J.R., Su, H., Jackson, D.R.: Free vibration of rotating tapered beams using the
dynamic stiffness method. J. Sound Vib. 298(4–5), 1034–1054 (2006)
21. Hsu, J.-C., Lai, H.-Y., Chen, C.-K.: Free vibration of nonuniform Euler-Bernoulli beams with
general elastically end constraints using Adomian modified decomposition method. J. Sound
Vib. 318(4–5), 965–981 (2008)
22. Huang, Y., Li, X.-F.: A new approach for free vibration of axially functionally graded beams
with non-uniform crosssection. J. Sound Vib. 329(11), 2291–2303 (2010)
23. Li, X.-F., Kang, Y.-A., Wu, J.-X.: Exact frequency equations of free vibration of exponentially
functionally graded beams. Appl. Acoust. 74(3), 413–420 (2013)
24. Mao, Q., Pietrzko, S.: Free vibration analysis of a type of tapered beams by using Adomian
decomposition method. Appl. Math. Comput. 219(6), 3264–271 (2012)
25. Ozgumus, O.-O., Kaya, M.O.: Flapwise bending vibration analysis of a rotating doubletapered Timoshenko beam. Arch. Appl. Mech. 78(5), 379–392 (2008)
26. Rajasekaran, S.: Free vibration of centrifugally stiffened axially functionally graded tapered
Timoshenko beams using differential transformation and quadrature methods. Appl. Math.
Model. 37(6), 4440–4463 (2013)
27. Rajasekaran, S., Norouzzadeh, T.E.: Free vibration analysis of axially functionally graded
tapered Timoshenko beams using differential transformation element method and differential
quadrature element method of lowest-order. Meccan. 49(4), 995–1009 (2014)
28. Yardimoglu, B.: Vibration analysis of rotating tapered Timoshenko beams by a new finite
element model. Shock Vib. 13(2), 117–126 (2006)
29. Baghani, M., Mazaheri, H., Salarieh, H.: Analysis of large amplitude free vibrations of
clamped tapered beams on a nonlinear elastic foundation. Appl. Math. Model. 38(3), 1176–
1186 (2013)
30. Huang, Y., Yang, L.E., Luo, Q.Z.: Free vibration of axially functionally graded Timoshenko
beams with non-uniform cross-section. Compos. Part B: Eng. 45, 1493–1498 (2013)
31. Shahba, A., Attarnejad, R., Hajilar, S.: A Mechanical-Based Solution for Axially Functionally
Graded Tapered Euler-Bernoulli Beams. Mech. Adv. Mater. Struct. 20, 696–707 (2012)
32. Asghari, M., Ahmadian, M.T., Kahrobaiyan, M.H., Rahaeifard, M.: On the size-dependent
behavior of functionally graded micro-beams. Mater. Des. 31, 2324–2329 (2010)
33. Asghari, M., Kahrobaiyan, M.H., Ahmadian, M.T.: A nonlinear Timoshenko beam formulation based on the modified couple stress theory. Int. J. Eng. Sci. 48, 1749–1766 (2010)
34. Lazopoulos, K.A., Lazopoulos, A.K.: Bending and buckling of thin strain gradient elastic
beams. Eur. J. Mech. A-Sol. 29, 837–843 (2010)